Solving $x^2 - 2x - 15 = 0$ Using The Zero Product Property

In the realm of mathematics, solving quadratic equations is a fundamental skill. Among the various techniques available, the Zero Product Property stands out as a particularly elegant and efficient method. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In this comprehensive guide, we will delve into the intricacies of the Zero Product Property and demonstrate its application in solving the quadratic equation $x^2 - 2x - 15 = 0$. We will embark on a step-by-step journey, meticulously dissecting each stage of the solution process to ensure clarity and comprehension. By the end of this exploration, you will not only be able to solve this specific equation but also possess a deep understanding of the Zero Product Property and its broader applications in solving quadratic equations.

Understanding the Zero Product Property

The bedrock of our approach lies in the Zero Product Property. This property, a cornerstone of algebra, asserts that if the product of two or more expressions equals zero, then at least one of the expressions must be zero. Mathematically, this can be expressed as follows: if $a imes b = 0$, then either $a = 0$ or $b = 0$ (or both). This seemingly simple principle forms the foundation for solving a wide range of equations, particularly quadratic equations that can be factored. The power of the Zero Product Property stems from its ability to transform a single equation into two or more simpler equations, each of which can be solved independently. This transformation simplifies the problem, making it more manageable and accessible. In the context of quadratic equations, the Zero Product Property allows us to break down a complex equation into linear factors, each of which can be easily solved to find the roots of the equation. Therefore, a thorough understanding of the Zero Product Property is crucial for anyone seeking to master the art of solving quadratic equations and other algebraic problems.

Step 1: Factoring the Quadratic Equation

The first crucial step in solving the quadratic equation $x^2 - 2x - 15 = 0$ using the Zero Product Property is to factor the quadratic expression. Factoring involves expressing the quadratic expression as a product of two linear factors. In this case, we seek two binomials that, when multiplied together, yield the original quadratic expression. To accomplish this, we need to identify two numbers that satisfy two key conditions: their product must equal the constant term (-15), and their sum must equal the coefficient of the linear term (-2). After careful consideration, we find that the numbers 3 and -5 fulfill these criteria, as $3 imes -5 = -15$ and $3 + (-5) = -2$. Therefore, we can rewrite the quadratic equation as $(x + 3)(x - 5) = 0$. This factorization is the cornerstone of our solution, as it transforms the original quadratic equation into a form where the Zero Product Property can be readily applied. By factoring the quadratic expression, we have effectively broken down the problem into simpler components, paving the way for a straightforward solution using the Zero Product Property.

Step 2: Applying the Zero Product Property

Now that we have successfully factored the quadratic equation into the form $(x + 3)(x - 5) = 0$, we can apply the Zero Product Property. As discussed earlier, this property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, the two factors are $(x + 3)$ and $(x - 5)$. Therefore, according to the Zero Product Property, either $(x + 3) = 0$ or $(x - 5) = 0$. This step is pivotal as it transforms the single quadratic equation into two separate linear equations, each of which can be solved independently. By applying the Zero Product Property, we have effectively decoupled the original problem into two simpler sub-problems. This strategic simplification is the key to unlocking the solution to the quadratic equation. Each of these linear equations represents a possible solution to the original quadratic equation, and solving them will reveal the values of $x$ that satisfy the equation.

Step 3: Solving the Linear Equations

With the Zero Product Property applied, we now have two simple linear equations to solve: $x + 3 = 0$ and $x - 5 = 0$. Solving these equations involves isolating the variable $x$ in each case. For the first equation, $x + 3 = 0$, we subtract 3 from both sides to obtain $x = -3$. For the second equation, $x - 5 = 0$, we add 5 to both sides to obtain $x = 5$. These two values of $x$, -3 and 5, are the solutions to the original quadratic equation $x^2 - 2x - 15 = 0$. Each value represents a point where the parabola defined by the quadratic equation intersects the x-axis. Therefore, solving these linear equations is the final step in determining the roots or solutions of the quadratic equation. By isolating $x$ in each equation, we have successfully identified the values that satisfy the original equation and represent the points where the quadratic function equals zero.

The Solution: $x = 5, -3$

Therefore, by applying the Zero Product Property and meticulously following each step, we have arrived at the solution to the quadratic equation $x^2 - 2x - 15 = 0$. The solutions are x = 5 and x = -3. This corresponds to option A in the given choices. These values of $x$ are the roots of the quadratic equation, meaning they are the values that make the equation true when substituted for $x$. They also represent the x-intercepts of the parabola defined by the equation $y = x^2 - 2x - 15$. Understanding the solutions to a quadratic equation is crucial in various mathematical and real-world applications, from modeling projectile motion to optimizing business processes. The Zero Product Property provides a powerful tool for finding these solutions, and mastering its application is an essential skill for anyone studying algebra and beyond.

Conclusion

In this detailed exploration, we have successfully solved the quadratic equation $x^2 - 2x - 15 = 0$ using the Zero Product Property. We began by understanding the fundamental principle of the Zero Product Property, which states that if the product of factors is zero, then at least one factor must be zero. We then systematically applied this property by first factoring the quadratic equation into $(x + 3)(x - 5) = 0$. Next, we set each factor equal to zero, resulting in two linear equations: $x + 3 = 0$ and $x - 5 = 0$. Solving these equations yielded the solutions $x = -3$ and $x = 5$. This process demonstrates the power and elegance of the Zero Product Property in solving quadratic equations. By breaking down the problem into smaller, more manageable steps, we were able to arrive at the solution in a clear and concise manner. The Zero Product Property is a valuable tool in the arsenal of any mathematician or student, and its mastery is essential for success in algebra and beyond. This step-by-step guide provides a solid foundation for understanding and applying the Zero Product Property to solve a wide range of quadratic equations.